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The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.

Consider randomly selecting such a coffee purchaser.

a. What is the probability that the individual purchased a small cup? A cup of decaf coffee?

b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability?

c. If we learn that the selected individual purchased decaf, what now is the probability that small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V).

a. What are the 16 outcomes in S?

b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate?

c. Which outcomes are in the event that all four mortgages are of the same type?

d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage?

e. What is the union of the events in parts (c) and (d), and what is the intersection of these two events?

f. What are the union and intersection of the two events in parts (b) and (c)?

A department store sells sports shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.

a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?

b. What is the probability that the next shirt sold is a medium print shirt?

c. What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?

d. What is the probability that the size of the next shirt sold is the medium? That the pattern of the next shirt sold as a print?

e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?

f. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?

According to July \(31, 2013\), posting on cnn.com subsequent to the death of a child who bit into a peanut, a \(2010\) study in the journal Pediatrics found that \(8\% \)of children younger than eighteen in the United States have at least one food allergy. Among those with food allergies, about \(39\% \)had a history of severe reaction.

a. If a child younger than eighteen is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction?

b. It was also reported that \(30\% \) of those with an allergy in fact are allergic to multiple foods. If a child younger than eighteen is randomly selected, what is the probability that he or she is allergic to multiple foods?

A system consists of two identical pumps, \(\# 1\) and \(\# 2\). If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, \(r\; = P\left( {\# 2\;fails|\# 1 fails} \right) > P\left( {\# 2 fails} \right) = q\). If at least one pump fails by the end of the pump design life in \(7\% \) of all systems and both pumps fail during that period in only 1%, what is the probability that pump \(\# 1\)will fail during the pump design life?

A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that \(P\left( A \right) = 0.6\)and\(P\left( B \right) = 0.05\). What is\(P\left( {B|A} \right)\)?

Given, \({A_i} = \) {awarded project i}, for \(i = 1, 2, 3\). Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one.

a. \(P\left( {{A_2}|{A_1}} \right)\)

b. \(P\left( {{A_2} \cap {A_3}|{A_1}} \right)\)

c. \(P\left( {{A_2} \cup {A_3}|{A_1}} \right)\)

d. \(P\left( {{A_1} \cap {A_2} \cap {A_3}|{A_1} \cup {A_2} \cup {A_3}} \right)\)

At a certain gas station, \({\rm{40\% }}\)of the customers use regular gas \(\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{,35\% }}\) use plus gas\(\left( {{{\rm{A}}_{\rm{2}}}} \right)\), and \({\rm{25\% }}\) use premium\(\left( {{{\rm{A}}_{\rm{3}}}} \right)\). Of those customers using regular gas, only \({\rm{30\% }}\) fill their tanks (event \({\rm{B}}\) ). Of those customers using plus, \({\rm{60\% }}\)fill their tanks, whereas of those using premium, \({\rm{50\% }}\)fill their tanks.

a. What is the probability that the next customer will request plus gas and fill the tank\(\left( {{{\rm{A}}_{\rm{2}}} \cap {\rm{B}}} \right)\)?

b. What is the probability that the next customer fills the tank?

c. If the next customer fills the tank, what is the probability that regular gas is requested? Plus? Premium?

A family consisting of three persons—A, B, and C—goes to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each memberof the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the station number for each member. One outcome is (1, 2, 1) for Ato station 1, Bto station 2, and Cto station 1.

a. List the 27 outcomes in the sample space.

b. List all outcomes in the event that all three members go to the same station.

c. List all outcomes in the event that all members go to different stations.

d. List all outcomes in the event that no one goes to station 2.

One of the assumptions underlying the theory of control charting is that successive plotted points are independent of one another. Each plotted point can signal either that a manufacturing process is operating correctly or that there is some sort of malfunction.